1. (a) If a chord of the parabola y? = 4a.x is a normal at one...

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1. (a) If a chord of the parabola y? = 4a.x is a normal at one of its ends, show that its mid-point lies on the curve y? 2(x – 2a) Sa Prove that the shortest length of such a chord is 6av3. [60 marks] (b) Find the asymptotes of the hyperbola r* - y? + 2r + y +9 = 0. [10 marks] 2. Verify that the point P(a cos 0, b sin 0) lies on the ellipse 2 y? 1, where a and b are the semi-major and semi-minor axes respectively of the ellipse. Find the gradient of the tangent to the curve at P and show that the equation of the normal at Pis ar sin 0 – by cos 0 = (a² - ) sin 0 cos 0. If P is not on the axes and if the normal at P passes through the point B(0,b), Show that a? > 26°. If further, the tangent at P meets the y-axis at Q, show that |BQ| [50 marks) EXAMINERS: T. Katsekpor, J. K. Ansong, P. K. Osei, J. Boiquaye, G. A. Botchwey Page 1 of 2 3. (a) Show that, for any complex number 2, 27 = |2/*, z + = 2Re(z) and Re(z) < |z|. Hence %3D show that i. |21 + 22 = |212 + |22? + 2Re(217). ii. |31 + z2| S |21| + |22l. where Re(2) is the real part of z and Z the conjugate of z. [26 marks] (b) If z1 = 1+2i, find the set of values of 22 for which (1) |31 + 2| = |21| + 22| (ii) |31 + za| = |2a| - 121. [50 marks] 1. (a) If a chord of the parabola y? = 4a.x is a normal at one of its ends, show that its mid-point lies on the curve y? 2(x – 2a) Sa Prove that the shortest length of such a chord is 6av3. [60 marks] (b) Find the asymptotes of the hyperbola r* - y? + 2r + y +9 = 0. [10 marks] 2. Verify that the point P(a cos 0, b sin 0) lies on the ellipse 2 y? 1, where a and b are the semi-major and semi-minor axes respectively of the ellipse. Find the gradient of the tangent to the curve at P and show that the equation of the normal at Pis ar sin 0 – by cos 0 = (a² - ) sin 0 cos 0. If P is not on the axes and if the normal at P passes through the point B(0,b), Show that a? > 26°. If further, the tangent at P meets the y-axis at Q, show that |BQ| [50 marks) EXAMINERS: T. Katsekpor, J. K. Ansong, P. K. Osei, J. Boiquaye, G. A. Botchwey Page 1 of 2 3. (a) Show that, for any complex number 2, 27 = |2/*, z + = 2Re(z) and Re(z) < |z|. Hence %3D show that i. |21 + 22 = |212 + |22? + 2Re(217). ii. |31 + z2| S |21| + |22l. where Re(2) is the real part of z and Z the conjugate of z. [26 marks] (b) If z1 = 1+2i, find the set of values of 22 for which (1) |31 + 2| = |21| + 22| (ii) |31 + za| = |2a| - 121. [50 marks]